De Moivre's Theorem and Certain Series which may be deduced from it 681 Value of Calculated by means of Gregory's Series - Euler's Series INTRODUCTORY CHAPTER ON THE THEORY OF DETEMINANTS Determinants of Three Rows - Determinants of Nine Elements Relation between Determinants of the Second and Third Orders Definition of the General Equation of the nth Degree - A Root. Definition of Cubic Equation-Cube Roots of Unity The Equation a3-a-Symmetrical Cubic Equation - Cubic Equation INTRODUCTION 1. In the study of Common Algebra the notion number is fundamental, and it is therefore necessary first of all to define it. What are the properties or characteristics of number? Given a group of objects, as marbles, a party of boys, a herd of horses, a village, or the like, distinctness or separateness of the things in any one of these groups is an intuitive property of these objects which enables one to realize that there is a marble, boy, horse, or house which is different and distinct from another marble, boy, horse, or house of the same group of marbles, boys, horses, or houses. If each marble of the group of marbles were replaced by an apple, then each apple by a nail, and so on; or if the marbles were painted different colors, arranged differently; or finally if any change were made in the things of the group which would not destroy their distinctness, the group of objects would contain as many individuals after any such change as it did before the change was made. The notion of number is based upon this property of the separateness of the things in a group, and is defined as that property of a group of different things which is unchanged no matter what change is made in the things of the group without destroying the distinctness of the individual things. Such changes affect only the character or arrangement of the things and do not cause any individual thing to be divided into two or more, or two or more to be merged into one. These characteristics of number expressed in the form of a theorem constitute the fundamental postulate of Arithmetic: The number of individual things in a group of things does not depend upon the order of their arrangement in the group, their characteristics, or the way they may be related to one another in smaller groups. |